Applications of Approximation Algorithms to
Cooperative Games
Kamal Jain
1. INTRODUCTION
The Internet, which is intrinsically a common playground
for a large number of players with varying degrees of collaborative and selsh motives, naturally gives rise to numerous
new game theoretic issues. Computational problems underlying solutions to these issues, achieving desirable economic
criteria, often turn out to be NP-hard. It is therefore natural to apply notions from the area of approximation algorithms to these problems. The connection is made more
meaningful by the fact that the two areas of game theory
and approximation algorithms share common methodology
{ both heavily use machinery from the theory of linear programming. Various aspects of this connection have been
explored recently by researchers [8, 10, 15, 20, 21, 26, 27,
29].
In this paper we will consider the problem of sharing the
cost of a jointly utilized facility in a \fair" manner. Consider
a service providing company whose set of possible customers,
also called users, is U . For each set S U C (S ) denotes the
cost incurred by the company to serve the users in S . The
function C is known as the cost function. For concreteness,
assume that the company broadcasts news of common interest, such as nancial news, on the net. Each user, i, has
a utility, u0i , for receiving the news. This utility u0i is known
only to user i. User i enjoys a benet of u0i xi if she gets the
news at the price xi . If she does not get the news then her
benet is 0. Each user is assumed to be selsh, and hence
in order to maximize benet, may misreport her utility as
some other number, say ui . For the rest of the discussion,
the utility of user i will mean the number ui .
A cost sharing mechanism determines which users receive
the broadcast and at what price. The mechanism is strategyproof if the dominant strategy of each user is to reveal the
Address: One Microsoft Way, Redmond, WA 98052, Email:
kamalj@microsoft.com.
y Address: College of Computing, Georgia Institute of Tech-
nology, Atlanta, GA 30332, Email: vazirani@cc.gatech.edu.
Research supported by NSF Grant CCR-9820896.
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STOC’01, July 6-8, 2001, Hersonissos, Crete, Greece.
Copyright 2001 ACM 1-58113-349-9/01/0007 ... 5.00.
$
Vijay V. Vaziraniy
true value of her utility. It is said to be group strategyproof
if this holds for coalitions as well. A cost sharing mechanism
is budget balanced if the total amount it charges from the receivers is same as the cost incurred by the service provider,
C (S ). It is ecient if it maximizes, over all subsets, S , the
sum of the utilities of users in S minus C (S ). Ideally, one
seeks an ecient, budget balanced and group strategyproof
cost sharing mechanism. A classical result in game theory
[13, 28] shows that such a mechanism does not exist even
for a submodular cost function (see Section 2 for formal definitions of these notions). Such a mechanism does not exist
even after relaxing the condition of group strategyproofness
to just strategyproofness.
In view of this limitation, there are two options: sacrice
either budget balance or eciency. In the rst case, one can
show that if the cost function is nondecreasing and submodular, then there is only one way of maximizing eciency [24].
This is called the marginal cost mechanism. This mechanism
is strategyproof, though not group strategyproof. It never
creates a budget surplus but can run a decit, and in many
cases raises no revenue at all [24].
In the second case, a fundamental theorem of Moulin and
Shenker [23, 24] shows that a cross-monotonic cost sharing method (also known as population monotonic allocation
scheme, see Section 2 for a denition) gives rise to a budget balance and group strategyproofness cost sharing mechanism. Moreover, if the cost function is submodular, the
converse holds as well.
A nondecreasing, submodular cost function supports an
entire class of cross-monotonic cost sharing methods. It
is useful to characterize methods from this class possessing special properties so that the service provider may pick
one judiciously. Two well known methods are:
1. Shapley Value [31]: In the context of multicasting
(see Section 2), this method distributes the cost of
each edge equally among all the users located downstream of the edge. Moulin and Shenker show that this
method achieves the lowest worst case loss of eciency
over all utility proles [24].
2. Egalitarian [5]: This method, due to Dutta and Ray,
seeks to distribute the cost equally among all the receivers. Mutuswami has shown that assuming all users
draw their utility values from the same probability distribution (with some technical restrictions) this method
maximizes the expected size of the set served [25].
A cost sharing method is said to satisfy the coalition participation constraint if the total cost share of any subset S
of the users is no more than C (S ), i.e., coalition S has no
incentive to break o from the grand coalition. only the
users in S . Such a cost sharing method is also called weakly
cross-monotone. The class of all weakly cross-monotone cost
sharing methods corresponding to a given cost function is
said to form the core. The classic Bondareva-Shapley Theorem states that the core is nonempty i the cost function
satises the covering property (see Section 5 for denition).
This allows us to compute an element of -approximate core
(see sections 4 and 5).
In this paper we rst consider the problem of multicast
routing. A tree T containing the source and all possible
users is xed. To serve a subset S of the users, information
is routed over the subtree of T containing S and the source.
Feigenbaum, Papadimitriou and Shenker [8] have recently
shown that the marginal cost cost-sharing mechanism can
be implemented on this model with the overhead of only
two messages per link, thus leading to a linear number of
total message, whereas Shapley value requires a quadratic
number of messages. They also show that the welfare value
of an optimal subtree is NP-hard to approximate within any
constant factor.
Although this model for multicast routing is widely used,
it suers from the drawback that the subtree connecting a
certain subset S of receivers may be arbitrarily more costly
than the cheapest tree to them. The latter of course is an
optimal Steiner tree containing S and the source. Such a
tree is NP-hard to compute. In addition, Megiddo [22] has
shown that for this game, the core is empty and so there does
not even exist a weak cross-monotonic cost sharing method
for the optimal Steiner tree.
We get around these diculties by turning to approximation algorithms. A well known factor 2 approximation
algorithm for Steiner tree is to nd a minimum spanning
tree on the required vertices [19]. The minimum spanning
tree game has been studied extensively in the literature [2,
11, 12, 17, 18, 16]. Kent and Skorin-Kapov [17] show how
to construct a whole class of weak cross-monotonic methods
as well as one cross-monotone method for this game. For
the latter result, they use matroid properties of spanning
trees. Using the primal-dual minimum spanning tree algorithm of Edmonds [6], we show how to construct a class of
cross-monotone methods. These methods are parameterized
by n mappings, fi : R+ ! R+ , one for each user, i. The
service provider now has a choice of methods. For instance,
he could use his estimates of the probability distribution
functions of users' utilities as the mappings.
In a forthcoming paper [14] we build on the above idea
of running a primal-dual-type algorithm and distributing
costs according to n mappings for constructing a class of
cross-monotonic cost sharing methods for an arbitrary nondecreasing submodular function. We also present a broader
notion of egalitarianism motivated by the following: If individual users have widely varying nancial resources then
the egalitarian method of Dutta and Ray does not equalize
their opportunity of receiving the service. We introduce the
notion of opportunity egalitarianism which attempts to do
this. We show how a service provider can attempt to equalize, among the users, the opportunity of their getting the
service, based on his estimate of their individual resources.
The basic technique underlying our method also enables us
to generalize Mutuswami's result: we give an algorithm that
maximizes the expected size of the set served even if indi-
vidual utilities are drawn from dierent probability distributions.
In Section 4 we point out a shortcoming of the condition
of budget balance, and propose the -approximate budget
balance condition. We use the cost sharing methods derived
for Steiner tree to give a 2-approximate budget balanced
group strategyproof cost sharing mechanism for the metric
TSP game (see Section 4), and leave the open problem of
deriving a factor 1.5 solution, using Christofedes' algorithm.
Core allocation functions for the metric and Euclidean TSP
games have been given by Faigle, Fekete, Hochstattler, and
Kern [7] and Fekete and Pulleyblank [9], respectively.
Several NP-hard minimization problems admit a constant
factor approximation algorithm based on an
LP-relaxation that is a covering program (e.g., see [32]). In
Section 5 we study general approaches for analyzing games
based on such problems. We show that the cost function
given by an optimal solution to a covering linear program
satises the covering property (see Section 5 for denitions).
We further show that it admits an eciently computable approximate weakly cross-monotonic cost sharing method.
Perhaps a stronger result holds { that in fact it admits
a cross-monotonic cost sharing method. We leave this as
an (important) open problem. If this open problem is settled positively, it will lead to a general scheme for obtaining a polynomial time computable -approximate budget
balanced group strategyproof cost sharing mechanism for
such games. Finding such a method for the facility location
game seems particularly interesting. Recently, Goemans and
Skutella [10] have studied this game and have a core allocation in case the well known LP-relaxation, due to Balinski
[1], has an integral solution.
Finally, we consider the purely combinatorial question of
characterizing the space of cross-monotone methods corresponding to a given nondecreasing submodular function. We
show that this is a polyhedron and characterize a subset of
its corner points. We leave the open problem of characterizing the rest of the corner points.
2. THE MODEL
Let us describe the model in the context of multicast routing. The simplest way of broadcasting a message to many
receivers is unicasting, under which the message is individually sent to each receiver. This results in several copies of
the same message traversing the network links. This waste
of bandwidth could be avoided by using a dierent form of
routing, called multicast routing [4]. Multicast routing uses
a tree connecting all the receivers to the source. The source
sends one copy of the message to each neighboring vertex in
the tree. These vertices further act as sources for the downstream receivers. In this manner, a source can reach many
receivers without sending multiple copies of a message over
any link. Traditionally, a tree containing the source and all
possible receivers is xed. The network is then restricted to
this tree only. Whenever a message needs to be broadcast
to a subset of receivers, multicast routing picks a subtree of
this tree. It therefore ignores the high connectivity oered
by today's Internet. This is not desirable when the receiver
set varies with the message, since the subtree to certain a
subset of receivers may be arbitrarily more costly than the
cheapest tree to them.
When the network is not restricted to a tree, the problem of nding the cheapest multicast tree for a given set
of receivers is the well-known Steiner tree problem, which
is NP-hard. Although constant factor approximation algorithms are known for the Steiner tree problem, we do not
know of any algorithm which gives either a nondecreasing
or a submodular cost function. In fact, the optimal solution
also does not give a submodular cost function. For example,
consider a four cycle with unit cost edges whose vertices are
three receivers and the source. The fractional optimal solution to the two well-known linear programming relaxations
of the Steiner tree problem, undirected cut formulation and
bidirected cut formulation, also does not give a submodular
cost function.
Let G = (V; E ) be an undirected graph with edge weights
ce 's and a marked node root, which is the broadcasting
source. All other nodes are users. We denote the set of
all users by U . We assume that a message can be duplicated at any node at no cost. Edge e charges the price ce
for transporting a message from one end to other. The cost
of broadcasting a message is the total price charged by all
the edges. We assume that each edge is of innite capacity,
so, a message can be sent from node i to node j through
the shortest path between them. Hence we assume that ce 's
satisfy the triangle inequality.
Now, suppose that the root has a message to broadcast
and every user i has reported utility ui to the root. The
root's job is to come up with the following:
1. a set Q of users, selected to receive the message,
2. a tree T containing Q to broadcast the message, and
3. for each user i, the price xi , to be charged as the cost
of delivering the message.
For notational convenience we will also represent Q by its
indicator function q, i.e., qi = 1 if i 2 Q and qi = 0 otherwise.
Now there are computational and economic constraints
to be satised. The only computational constraint we are
considering in this paper is that only polynomial time is
available for computation. The economic constraints are
listed below:
1. Optimality. T is an optimum tree connecting all the
users in Q with the root.
2. No Positive Transfers (NPT). For each user i, xi 0, i.e., users will not be paid for receiving a message.
3. Voluntary Participation (VP). qi ui xi 0 i.e.,
if i 62 Q then xi = 0 and if i 2 Q then xi ui i.e.,
only those users will pay who will receive the message.
Moreover, they will never be asked to pay more than
their reported utilities. In other words each user has
the option to not receive the message, and if so, derives
a benet of 0.
4. Consumer Sovereignty (CS). Every user is guaranteed to receive the message if she reports a high enough
utility value.
5. Budget-Balance (BB).
P
P
(a) Cost Recovery i2Q xi e2T ce , i.e., the
cost of broadcasting the message is recovered from
all the users.
P
P
(b) Competitiveness i2Q xi 6 e2T ce , i.e., no
surplus is created. Because if any surplus is created then a competitor can deliver the message
at a cheaper cost by reducing the surplus.
The condition of budget balance consists of P
satisfying
cost recovery and competitiveness, i.e., i2Q xi =
Pboth,
e2T ce (the set of users receiving the message pay exactly the total cost of T ).
P
Pe2T ce is maximized, i.e., as
6. Eciency. i2Q ui
much worth is created
we want
P as possible.
Pe2TIdeally
Pareto optimal, i.e., i2Q u0i
ce to be maximized. But the next condition reduces the latter to
the former.
7. Group Strategyproofness. Even if a set of users
collude, the dominant strategy for each player is to
report her utility value truthfully. To be precise, consider a coalition C of users. Let uj = u0j for all j 62 C .
Let (q; x) and (q0 ; x0 ) denote the users served and costs
at u and u0 respectively. Now, group strategyproofness
requires that if the inequality
u0i qi xi u0i qi0 x0i
holds for all i 2 C then it must hold with equality
for all i 2 C as well, i.e., if no member of C is made
worse o by misreporting their utility values then no
member of C is made better o either.
Satisfying the rst condition is not computationally possible, assuming P 6= NP. We relax it to: the cost of T is
within twice the cost of the optimum tree connecting users in
Q with the root. A classical result in game theory shows that
there is no strategyproof mechanisms that is both budget
balance and ecient. Furthermore, Feigenbaum, Papadimitriou and Shenker have shown that it is NP-hard to compute
a constant factor approximation to condition 6 as well. So,
we are going to put this condition aside. Moulin and Shenker
showed that all other economic constraints mentioned above
can be captured by a cross-monotonic cost sharing method.
A cost function is submodular if
1. C (;) = 0,
2. for any Q1 and Q2 , C (Q1 ) + C (Q2 ) C (Q1 [ Q2 ) +
C (Q1 \ Q2 ).
It is supermodular if the inequality in the second condition
is reversed. A cost function is nondecreasing if for Q1 Q2 ,
C (Q1 ) C (Q2 ).
A cost allocation function f distributes the cost of sending
the message
U; f (i) 0
P to the entire set of users U , i.e.,of8alli 2cost
and i2U f (i) = C (U ). The core consists
allocaP
tion functions f such that 8S U; i2S f (i) C (S ), i.e.,
no subset of the users have an incentive to secede.
A cost sharing method is a function, , which distributes
the cost of broadcasting the message to the recipients. More
formally, takes two arguments, a set of users Q and a
user i, and returns a nonnegative real number satisfying the
following:
1. if i 62 Q then (Q; i) = 0 and
P
2. i2Q (Q; i) = C (Q), where C (Q) represents the cost
of broadcasting a message to Q. Note that C (Q) is
not the optimum cost but it is the cost of the tree T
computed for broadcasting the message.
A cost sharing method is cross-monotonic if for Q R,
(Q; i) (R; i) for everyPuser i 2 Q. ItPis weakly crossmonotonic if for Q R, i2Q (Q;i) i2Q (R; i): As
the name suggests, a cross-monotonic method is weakly crossmonotonic as well. Observe that a weakly cross-monotonic
method provides us with a core allocation for each subset of
the users.
For every cross-monotonic cross sharing method, , Moulin
and Shenker give the following mechanism M ( ) which computes Q and xi = (Q;i).
Mechanism M ( )
1. Q is initialized to U .
2. if there is a user i in Q with ui < (Q; i) then drop i
from Q. Keep repeating this step, in arbitrary order,
until for every user i in Q, ui (Q;i).
3. set xi = (Q;i).
This mechanism starts with an attempt to send message
to all users, and uses to determine the cost share of each
user. It drops, in arbitrary order, any user who cannot pay
for receiving the message. The mechanism iterates until it
a set of users who can together pay for the cost of the tree
needed to serve them. Because of cross-monotonicity of ,
the eventual set of users left does not depend on the order
in which individual users are dropped. Since in every step
at least one user is dropped, the mechanism will stop in a
linear number of iteration.
Theorem 1. [24, 23] For any cross monotonic cost sharing method , the mechanism M ( ) is budget balanced, meets
NPT, VP, CS and is group strategyproof.
As shown in [23, 24], the converse of Theorem 1 holds
for any submodular function. For completeness, we provide
a (slightly simpler) proof of group strategyproofness. All
other conditions are easy to verify.
Proof. We will show that M ( ) is group strategyproof,
for a cross monotonic cost sharing method . Let u0 be
the true utility prole. Assume a coalition C manipulates
at u0 by u (where u0i = ui for all i 62 C ). Consider two
runs, R(u0 ) and R(u) of the above mechanism with true and
false utility values respectively. Let Q0 and Q be the output
sets produced by the two runs, and q0 and q be the vectors
representing these sets. Assume further that for each i 2 C ,
the inequality
u0i qi xi u0i qi0
x0i
(1)
holds. We will show that for each i 2 C , this inequality
must hold with equality, thereby showing that M ( ) is group
strategyproof.
In an iteration of M ( ), there could be several users, i,
satisfying ui < (Q; i). Since is cross monotonic, the nal outcome does not depend on the manner in which a
user is picked to be dropped. Each of the runs R(u0 ) and
R(u) makes these choices in an arbitrary (and uncorrelated)
manner. Let s1 ; s2 ; : : : ; sk be the order in which users were
dropped in run R(u0 ).
We will rst prove, by contradiction, that Q Q0 . Let
si be the rst user in the list s1 ; s2 ; : : : ; sk such that si 2
Q. By the choice of i, each of the users s1 ; : : : ; si 1 has
been dropped in run R(u), and so U fs1 ; : : : ; si 1 g Q.
Now, by cross monotonicity of , (U fs1 ; : : : ; si 1 g; si ) (Q; si ). Since si is dropped in run R(u0 ) and retained in run
R(u), u0si < (U fs1 ; : : : ; si 1 g; si ), and (Q; si ) usi .
Now, there are two cases. If si 2= C , by assumption, u0si =
usi . But the above inequalities give u0si < usi , thereby
leading to a contradiction. The second case is that si 2 C .
Since qs0 i = 0 and qsi = 1, by (1), u0si (Q; si ). But the
above inequalities give u0si < (Q; si ), again leading to a
contradiction. Hence, Q Q0 .
Now, by cross monotonicity of , the price paid by any
member of C in the second run must be at least the price
paid in the rst run, i.e., 8i 2 C; (Q; i) (Q0 ; i). Hence,
for each i 2 C , (1) must hold with equality.
3. THE STEINER TREE GAME
In this section, we will give a class of cost-sharing methods
for multicasting that achieves budget balance and is crossmonotone, and for any set Q of users chosen for distribution,
nds a tree of cost at most twice the optimal Steiner tree
containing the root and users Q. Our method utilizes the
following two well known facts.
If the edge costs satisfy the triangle inequality, the
cost of a minimum spanning tree on the set of required
vertices is within twice the cost of an optimal Steiner
tree containing all required vertices.
There is an exact linear programming relaxation for
the minimum spanning tree problem, i.e., a relaxation
that always has optimal integral solutions.
The rst fact is due to [19]. The second follows from a
more general fact due to Edmonds: that there is an exact relaxation for the minimum branching problem. In this problem, we are given a directed graph with nonnegative costs on
the directed edges, and one of the vertices is marked as root.
The problem is to nd a minimum cost tree containing all
vertices and directed into the root. The transformation from
the minimum spanning tree problem in an undirected graph
to the minimum branching problem is straightforward. Simply replace each undirected edge e = (u; v) of G by two
directed edges (u ! v) and (v ! u) each of cost ce , and
ask for a minimum cost branching directed into the root.
Let us denote this directed graph by H = (V; E~ ). Once the
branching is found, we ignore directions on edges to obtain
a minimum spanning tree in G.
Let us say that a set S V is valid if it is nonempty
and does not contain the root. For any set S V and
F E~ let (S ) = f(u ! v) 2 E~ j u 2 S and v 2 S g, and
F (S ) = f(u ! v) 2 F j u 2 S and v 2 S g. We state below
the LP-relaxation and dual.
minimize
subject to
Xc x
2
(2)
e e
X
~
e E
2
e: e (S )
xe 0;
xe 1 ;
8 valid set S
e 2 E~
maximize
subject to
X
valid set S
X
2
S : e (S )
y S ce
yS 0
(3)
yS
e 2 E~
8 valid set S
Let us say that edge e feels dual yS if yS > 0 and e 2 (S ).
Say that edge e is tight if the total amount of dual it feels
equals its cost. The dual program is trying to maximize the
sum of the dual variables yS subject to the condition that no
edge feels more dual than its cost, i.e., no edge is over-tight.
We present Edmonds' algorithm below, which is based on
the primal-dual schema. Starting with the trivial primal
and dual solutions, it iteratively improves the feasibility of
the primal and the optimality of the dual. When an edge
becomes tight, it is included in the ordered list F . In any
iteration, a set S V is said to be unsatised if it is valid
and F (S ) = ;. Any minimal unsatised set is said to be
active. It is easy to see that active sets must be disjoint and
must be strongly connected w.r.t. F .
For the purpose of proving properties of this algorithm,
we will associate a notion of time with the algorithm. In
unit time, the algorithm grows duals a unit amount. In an
iteration, the algorithm nds all active sets, and raises their
dual variables until a new edge, say e, goes tight. At this
point, the current iteration ends, and edge e is appended to
the list F . The algorithm continues until there are no more
unsatised sets. At this point, the algorithm prunes F using
the procedure of reverse delete. The edges that remain in F
form a branching directed into the root.
Algorithm 2 (Minimum branching)
1. (Initialization) F ;; for each S V , yS 0.
2. (Edge augmentation) While there exists an
unsatised set do:
Find all active sets w.r.t. F. For each such set S,
raise its dual variable yS until some edge e goes
tight;
F F [ feg.
3. Let e1 ; e2 ; : : : ; el be the ordered list of edges in F.
4. (Reverse delete) For j = l downto 1 do:
If there are no unsatised sets w.r.t. F fej g, then
F F fej g.
5. Return F .
Because the algorithm raises dual variables only for strongly
connected sets, and does a reverse delete in the end, it is possible to show that every valid dual S such that yS > 0, must
satisfy jF (S )j = 1. This, and the fact that only tight edges
are picked lead to showing that the cost of the branching
picked is precisely equal to the total dual raised, i.e.,
Xc
2
e F
e
=
X
valid set S
yS :
This important fact shows that the branching found by
the algorithm is optimal. As a consequence, we get that
LP (2) always has an integral optimal solution. This fact
will also enable us to show that our cost-sharing method is
budget balanced.
Let Q be the set of recipients. We are provided with
functions fi : R+ ! R+ , one for each user i. Use Algorithm
2 to nd a minimum spanning tree containing Q and the
root.
We now dene the cost-sharing method . The cost share
for user i 2 Q is computed as follows. Let T denote the rst
time at which there is a path from i to the root consisting
of tight edges. At time t T , let S (t) denote the set of
vertices reachable from i using tight edges. At each time
t < T , the algorithm grows the dual variable yS(t) at unit
rate. Let
F (t) =
X f (t):
j
2
j S (t)
Dene the cost share for user i,
(Q; i) =
Z
T
0
fi (t) dt:
F (t)
Theorem 3. The cost-sharing method, , is cross-monotonic.
Proof. Observe that at each time, the cost-sharing method
is simply distributing the growing dual yS(t) among the users
in S (t). Since the total dual constructed equals the cost of
the tree found,
X (Q;i) = C (Q):
2
i Q
Consider a run of Algorithm 2 on the set of users Q and
let w 2= Q be another user. Let i 2 Q be an arbitrary user.
Let R and R0 be the runs of Algorithm 2 on input Q and
Q [ fwg, respectively. Let S (t) and S 0 (t) be the sets of
vertices reachable from i via tight edges at time t in runs R
and R0 , respectively. Dene
F (t) =
X f (t)
and F 0 (t) =
j
2
j S (t)
X
2
fj (t):
j S 0 (t)
Let T and T 0 be the rst times at which there is a tight path
from i to the root in run R and R0 , respectively. Then,
(Q; i) =
Z
T
0
fi (t) dt;
F (t)
and
(Q [ fwg; i) =
Z
0
T
0
fi (t)
F 0 (t) dt:
If i can reach w in run R0 at time t, then S 0 (t) S (t), and
otherwise S (t) = S 0 (t). Therefore, for all t, F 0 (t) F (t).
Furthermore, i reaches the root at the same time or earlier
in run R0 than in run R, i.e., T 0 T . Hence, (Q [fwg;i) (Q; i).
For purposes of eciency, Algorithm 2 is rst run to determine the duals grown, and for each dual, the time at which
it started and stopped growing. Since the duals grown form
a laminar family, their number is bounded by 2n. This information is sucient for cost allocation. If the functions fi
are simple, the cost shares can be computed in closed form;
otherwise, one will have to use numerical methods.
Corollary 4. For the cost sharing method given above,
the mechanism M ( ) is budget balanced, meets NPT, VP,
CS, and is group strategyproof.
If all the functions fi are the same, we get the cost-sharing
method of Kent and Skorin-Kapov. This corresponds to
distributing each dual yS equally among the users in the set
S.
Are all possible cross-monotone cost sharing methods for
minimum spanning trees captured by the algorithm given
above? We give an example to show that the answer to this
question is \No". Consider a network on 3 vertices, root, u
and v. Let the distances be (root;u) = 1, (u; v) = 1 and
(root; v) = 2. Consider the cross-monotone cost-sharing
method (fug; u) = 1, (fvg; v) = 2, (fu; vg; u) = 0 and
(fu; vg; v) = 2. When run with vertices u and v, our algorithm will assign a nonnegative cost to vertex u. Hence, it
will never generate this cost-sharing method.
4. RELAXING THE
BUDGET BALANCE CONDITION
Clearly, the cost sharing method should be such that the
service provider does not run into a decit. In the presence
of competition, it should not create a large surplus either.
Budget balance ensures both these conditions, in a mathematically clean manner. However, this is a dicult condition to ensure. Moreover, even when satised, it can suer
from the following aw. Let us consider our Steiner tree solution. We assumed that we were provided with a complete
graph, G, with edge costs satisfying the triangle inequality,
and found a tree T in it. However, the original network, H ,
may not have links connecting all pairs of nodes. Graph G
is obtained by taking the closure of H , so that edges in G
correspond to shortest paths in H . Mapping tree T back
to H involves replacing edges by paths. This gives rise to a
spanning graph, say H 0 , which in general contains cycles and
multiple edges. Now, budget balance requires the broadcasting company to send a message on each link of H 0 , which
is clearly wasteful. Any spanning tree that is a subgraph of
H 0 , say T 0 , suces.
This aw and the general diculty of ensuring budget balance motivate the following denition. Let OPT(Q) denote
the optimal cost function for serving users Q and let 1
be a constant. (In general, may be a function of n { for this
paper, let us assume it is a constant.) is an -approximate
cost sharing method if it satises the following:
1. -approximate
Competitiveness
8Q U : Pi2Q (Q; i) OPT(Q), i.e., any set Q
of users together pays at most times the optimum
cost of serving Q.
P
2. Cost Recovery 8Q U : i2Q (Q; i) C (Q),
for some feasible solution of cost C (Q), i.e., the cost
incurred by the service provider is recovered from the
users served.
3. if i 62 Q then (Q; i) = 0.
is cross-monotonic if for Q R, (Q; i) (R; i) for
user i 2 Q. P
It is weakly cross-monotonic if for Q R,
Pevery
i2Q (Q; i) i2Q (R; i). It is said to be eciently
computable if for any Q, both C and are polynomial time
computable.
A cost sharing mechanism is -approximate budget balanced if it is -approximate competitive and cost recovering.
One can easily obtain the following along the lines of Moulin
and Shenker's Theorem 1.
Theorem 5. For any -approximate cross-monotonic cost
sharing method , mechanism M ( ) is -approximate budget balanced, meets NPT, VP, CS and is group strategyproof.
Furthermore M ( ) is eciently computable if is.
Under these denitions, tree T 0 is a 2-approximate budget balanced solution. Let us illustrate another use of this
notion. Consider the metric TSP game, in which edge costs
between nodes satisfy the triangle inequality, and a traveling salesman starts at node 1 and executes a tour, visiting a
set of users that are chosen by the cost sharing mechanism.
Recall that doubling an MST, nding an Eulerian tour and
short cutting gives a factor 2 approximation algorithm for
metric TSP. Using this fact and Theorem 3 we get.
Theorem 6. There is a 2-approximate budget balanced
group strategyproof cost sharing mechanism for the metric
TSP game.
An -approximate cost allocation function f is an
-approximate competitive and cost recovering way of serving the entirePset of users U , i.e., 8i 2 U; f (i) 0 and
OPT(U ) i2U f (i) C (U ). The -core consists of all
-approximate
cost allocation functions f such that 8S P
U; i2S f (i) OPT(S ). Clearly, an -approximate weakly
cross-monotonic cost sharing method yields an -approximate
cost allocation function for each subset of users. A shortcoming of the notion of core is that it turns out to be empty
for many games. We hope that the relaxed notion introduced above will alleviate this diculty.
Remark : In the denition of -approximate weakly crossmonotonic
sharing
Pi2Q (Q; icost
P method, replacing
P the condition
) i2Q (R;i) by i2Q (R; i) OPT(Q)
would have lead to a weaker, though still potentially useful,
denition.
5. A GENERAL APPROACH
Next, we present an open problem, whose positive resolution would lead to a general technique for obtaining an approximate budget balanced group strategyproof cost sharing method for several games based on NP-hard problems.
For the time being, we can construct -approximate core
allocations for these games. Let us rst present some denitions.
A fractional set, S , is a set in which an element can appear
partially i.e., with each element, e, there is a number fS (e) 2
[0; 1], which tells the extent of appearance of e in set S . The
union of two fractional sets, S1 and S2 , is denoted by S1 [ S2
and is dened by the function fS1 [S2 = minffS1 + fS2 ; 1g: If
S is a set and f 2 [0; 1] then f S is a fractional set where each
element in S appears to the extent of f in f S . Fractional
sets S1 ; S2 ; : : : ; Sn cover S if S1 [ S2 [ : : : [ Sn = S .
Suppose U is the set of all users and C : 2U ! R+ is
a cost function. C is said to exhibit the covering property
if forSany set S of users and any P
covering of S of the form
S = j fj Sj , we have C (S ) j fj C (Sj ), where each
Sj is a set of users.
The classic Bondareva-Shapley Theorem [3, 30] shows that
a necessary and sucient condition for the existence of a
weakly cross-monotonic cost sharing method is that the underlying cost function exhibit the covering property.
Does a cost function satisfying the covering property always admit a cross-monotonic cost sharing method? A positive resolution will lead to the following general scheme.
A covering linear program is a minimization linear program in which all coecients in the constraint matrix and
the objective function are nonnegative. Let L be a covering
linear program. The feasible region of homogenous linear
inequalities is called a cone. A cone that lies in the nonnegative orthant will be called a nonnegative cone. Consider
the feasible solutions of L that lie in a nonnegative cone
C . Corresponding to each user is a set of constraints of the
linear program. For instance, in case of the minimum spanning tree LP, the constraints corresponding to a user i corresponds to sets S V containing i but not the root. Each
user has a utility value to get the corresponding constraints
satised. This utility value can be misreported. Let Ci be
the set of constraints corresponding to a user Si. The set of
constraints corresponding to the users in Q is i2Q Ci . The
cost, C (Q), for serving Q is the optimum objective function
S
value of a solution that satises all constraints in i2Q Ci
and lies in C . Without loss of generality we assume that
each set of constraints corresponds to a user.
Lemma 7. For any covering LP L and nonnegative cone
C , the cost function as dened above exhibits the covering
property.
Sf
j 's are
j j Sj , where S and SP
sets of users. We want to establish that C (S ) j fj C (Sj ). Consider the right hand side. Let xj be an optimal
solution for satisfying
and lying in
P all the constraints inC Sasj well.
cone C . Clearly, j fj xj lies in the cone
Hence,
P
the lemma is proven if we show that j fj xj serves all the
users in S . Consider a user i in S . User i is served if all
the constraints in Ci are satised. Consider a constraint, say
aT x b, in Ci . Note that there is no negative coecient
in a or b. Also P
we have restricted
x to nonnegative
C.
Pj:i2Scone
T
T P
T
Therefore,
a
f
f
x
=
f
j xj a ja j
j
:i2Sj j j
j
xj Pj:i2Sj fj b b, where theSlast inequality follows from
that i is covered in j fj Sj . This shows that
Pthej ffact
j xj satises all the users in S . The cost of the optimum
way to satisfy all users in S , C (S ), can only be smaller.
Proof. Suppose S =
Consider an NP-hard minimization problem for which
an factor approximation algorithm is obtained using an
LP-relaxation as a lower bound, i.e., the cost of the solution found is at most times an optimal solution to the
LP. Furthermore, assume that this LP, P , is a covering LP,
L, intersected with a nonnegative cone C . (In most cases C
will simply be the nonnegative orthant. As shown below, for
facility location, a dierent cone is required.) If our open
problem resolves positively, the optimal cost function for LP
P admits a cross-monotonic cost sharing method. Multiplying by gives us an -approximate budget balanced group
strategyproof cost sharing mechanism.
Using Lemma 7 and the Bondareva-Shapley Theorem [3,
30] one can show that there exists an -approximate weakly
cross monotonic cost sharing method for . However, their
theorem uses an exponential sized LP which may be solvable in polynomial time in particular cases, though not in
general. The following theorem gives a way of nding one
such method eciently.
Theorem 8. There is an eciently computable
-approximate weakly cross-monotonic cost sharing method
for .
Proof. We may assume w.l.o.g. that there is a unique
inequality in L corresponding to each user and that the constant in this inequality is 1. The latter is easily ensured by
scaling. Suppose the former is not satised for user i. Consider the inequalities corresponding to user i. Pick a new
variable xi and replace each inequality aT x b by the
homogenous inequality aT x bxi . These homogenous inequalities can be pushed into the cone. Further, add the
inequality xi 1 in L. Clearly the new LP is equivalent to
the old one.
Let D be the dual linear program for P . For subset Q of
users, let P Q be the restriction of P to inequalities corresponding to Q only (of course retaining all of C ). Let DQ
denote the dual of P Q .
Consider the following cost sharing method. Let yQ denote an optimal solution to LP DQ . The cost share of user
i 2 Q is (Q; i) = yQ (i). If i 2= Q, then (Q; i) = 0. Let
OPTf (Q) denote the objective function value of solution
yQ .
This cost
P sharing method is -approximate competitive
because i2Q (Q; i) = OPTf (Q) OPT(Q). It satises cost recovery because the cost of the solution produced by the factorPapproximation algorithm for is
C (Q) OPTf (Q) = i2Q (Q; i).
Finally, let us show that it satises weak cross-monotonicity.
Let Q R. Let yR be an optimal solution to LP DR . Let
y0 denote the restriction of yR to coordinates correspond-0
ing to users in Q. The important observation is that y
is a feasible solution to DQ . Since the optimal solution to
DQ can have only a higher objective function value, weak
cross-monotonicity follows.
Let us show how Lemma 7 helps overcome some of the
diculties in obtaining a group strategyproof cost sharing
method for the facility location game. In this game, we are
given a set of cities and a set of potential sites for opening
facilities. For each site, we are given the cost of opening
a facility there. For each city and site we are given the
cost of connecting the city to a facility opened at that site.
These connection costs satisfy the triangle inequality. In this
setting cities are users who want themselves to be connected
to an open facility. Each user reports a utility for being
connected to an open facility. The cost of serving a set of
users is the total cost of opening facilities and connecting
each city in the set to one of the open facilities.
One diculty is that the optimal solution does not admit
a weakly cross-monotonic cost sharing method (and hence
no cross-monotonic method either). The reason is that the
optimal cost function does not exhibit the covering property.
For an instance, consider a cycle on 6 vertices, with 3 cities
and 3 facilities alternating. The cost of each edge is 1 and
the cost of opening each facility is 2.
A well known LP-relaxation for this problem, due to Balinski [1], is given below. Several constant factor approximation
algorithms are based on this relaxation. Suppose C is the
set of cities and F is the set of facilities. Suppose fi is the
cost of opening facility i and cij is the cost of connecting
city j to facility i. In the corresponding integer program, yi
is a 0=1 variable which is 1 i facility i is open, and xij be
a 0=1 variable which is 1 i city j is connected to facility i.
min
(S; i) = C (S ) C (S fig):
Proof. By the cross-monotonicity of , max (S; i) C (i). Furthermore, this value is attained for the incremental
cost-sharing method based on any permutation such that
(1) = i.
Applying cross-monotonicity to S fig, we get that
min
(S; i) C (S ) C (S fig):
This value is attained for the incremental cost-sharing method
(4) based on any permutation such that (n) = i.
cij xij
minimize
i i
i2F
j 2C;i2F
This leads to the following question regarding the miniX
mum spanning tree game: what is the complexity of comsubject to
xij 1;
8j 2 C
puting the minimum and maximum, over all cross-monotone
i2F
cost-sharing methods, of the cost-share of an individual user
yi xij 0;
8j 2 C; 8i 2 F in a given coalition? This problem appears to be NP-hard.
Let be a cross-monotone cost-sharing method for C .
xij 0;
i 2 F;j 2 C
Represent as a point in n2n 1 dimensional real space
yi 0;
i2F
whose coordinates are indexed by pairs (Q; i), where Q U
and i 2 Q. Consider in this space the set of all crossmonotone cost-sharing methods for C . This set is a polyThis LP has negative coecients and so it is not a covtope, since it can be described by the following linear system
ering LP. Note that last three sets of constraints dene a
on n2n 1 variables x(Q; i) where Q U and i 2 Q.
nonnegative cone. Let us call it C . We can rewrite this LP
as follows:
X x(Q; i) = C (Q); 8Q U
X
X
i2Q
minimize
cij xij + fi yi
(5)
j 2C;i2F
i2F
x(Q; i) x(Q0 ; i);
8Q0 U; Q Q0 ; i 2 Q
X xij 1;
x(Q; i) 0;
8Q U; i 2 Q
8j 2 C
subject to
i2F
Proposition 10. Each incremental cost sharing method
C
for C is a corner points of the polytope described above.
Therefore, by Lemma 7, the optimal solution to BalinProof. Suppose is not a corner point of the polyski's LP satises the covering property. This opens the
tope. In this case it can be written as the convex compossibility of nding a constant factor approximate budget
balanced and group strategyproof method for the facility
bination of two other cross-monotonic cost-sharing methods, a and b . We claim that a , b and are the same.
location game.
Suppose not. Then there exists Q U and (i) 2 Q
such that a (Q; (i)) 6= (Q; (i)) 6= b (Q; (i)). Let us
6. CHARACTERIZING
pick the smallest such i. Without loss of generality asCROSS-MONOTONIC METHODS
sume that a (Q; (i)) < (Q; (i)) < b(Q; (i)). Let
Q0 = f(j ) : j ig \ Q. Since is an incremental costLet C be a nondecreasing submodular cost function over
sharing method,
the user set U = f1; 2; : : : ; ng. In this section we will study
the space of all cross-monotone cost-sharing methods for C
X (Q; (j)) = C (Q0):
as well as the smallest and largest cost shares allocated by
such methods to a specic user in a specic coalition.
(j )2Q
We rst dene some special cost sharing methods. Let be a permutation on 1; : : : ; n. The incremental cost sharing
Therefore,
method is dened as follows. Let S U . Assume jS j = k,
X b(Q; (j)) > C (Q0):
and let i1 ; : : : ; ik be the users in S , ordered according to .
The cost shares assigned to these users by are: (S; i1 ) =
(j )2Q
C (i1 ), and for 2 j k, (S; ij ) = C (fi1 ; : : : ; ij g)
C (fi1 ; : : : ; ij 1 g).
This contradicts the cross-monotonicity of b .
Proposition 9. Let S U and i 2 S . Among all crossDo all the corner points of the polytope correspond to inmonotone cost-sharing methods for C ,
cremental cost-sharing methods? The following cross-monotone
cost-sharing method cannot be written as a convex combinamax
(S; i) = C (i) and
tion of incremental cost-sharing methods, thereby showing
X
X
+ fy
0
0
that the answer to this question is \No". (fa; b; cg; a) =
2, (fa; b; cg;b) = 3, (fa; b; cg; c) = 4, (fa; bg; a) = 4,
(fa; bg; b) = 3, (fa; cg; a) = 3, (fa; cg; c) = 4, (fb; cg; b) =
3, (fb; cg; c) = 4, (fag; a) = 4, (fbg; b) = 4, (fcg;c) = 4.
Observe that the cost function of this example is particularly simple: it assigns costs to sets based only on their
cardinality. Let us call such a cost function a cardinality
cost function.
We leave the open problem of characterizing the rest of
the corner points of this polytope. The special case of a
cardinality cost function is also interesting.
7. ACKNOWLEDGEMENTS
We wish to thank Daniel Granot, Herve Moulin, Suresh
Mutuswami, Christos Papadimitriou, and Scott Shenker for
valuable comments and for providing pointers into the literature.
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